Download App

Differential Equations — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
Question
Solve the following differential equation
$\text{xy}(\text{y}+1)\text{dy}=(\text{x}^2+1)\text{dx}$
✓

Answer

We have $\text{xy}(\text{y}+1)\text{dy}=(\text{x}^2+1)\text{dx}$$\Rightarrow\{\text{y}(\text{y}+1)\}\text{dy}=\frac{\text{x}^2+1}{\text{x}}\ \text{dx}$
$\Rightarrow(\text{y}^2+\text{y})\text{dy}=\Big(\text{x}+\frac{1}{\text{x}}\Big)\text{dx}$
Integrating both sides, we get
$\int(\text{y}^2+\text{y})\text{dy}=\int\Big(\text{x}+\frac{1}{\text{x}}\Big)\text{dx}$ $=\int\text{y}^2\text{dy}+\int\text{y dy}=\int\text{x dx}+\int\frac{1}{\text{x}}\text{ dx}$ $\Rightarrow\frac{\text{y}^3}{3}+\frac{\text{y}^3}{2}=\frac{\text{x}^2}{2}+\log|\text{x}|+\text{C}$ Hence, $\frac{\text{y}^3}{3}+\frac{\text{y}^3}{2}=\frac{\text{x}^2}{2}+\log|\text{x}|+\text{C}$ is the required solution.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(5 Marks)" from Differential Equations→Differential Equations — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

using interation, find the area of the region bounded by the triangle ragion, the euations of whose sides are$ y = 2x + 1, y = 3x + 1$ and $x = 4.$
→
Determine the values of a, b, c for which the function
$\text{f}\text{(x)}=\begin{cases}\frac{\sin\text{(a}+1)\text{x}+\sin\text{x}}{\text{x}}, &\text{for}\text{ x}<0,&\\\text{ c},&\text{for x}=0\\\frac{\sqrt{\text{x}+\text{bx}^2}-\sqrt{\text{x}}}{\text{bx}^\frac{3}{2}},&\text{for x}>0\end{cases}$ is continuous at x = 0.
→
An article manufactured by a company consists of two parts X and Y. In the process of manufacture of the part X, 9 out of 100 parts may be defective. Similarly, 5 out of 100 are likely to be defective in the manufacture of part Y. Calculate the probability that the assembled product will not be defective.
→
Show that the following system of linear equation is inconsistent:
$2x + 5y = 7$
$6x + 15y = 13$
→
Evaluate the following integrals:
$\int\text{e}^{2\text{x}}\cos^2\text{x }\text{dx}$
→
A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman's time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftman's time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftman's time.
  1. What number of rackets and bats must be made if the factory is to work at full capacity?
  2. If the profit on a racket and on a bat is Rs. 20 and Rs. 10 respectively, find the maximum profit of the factory when it works at full capacity.
→
Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=\text{x}^{\cos\text{x}}+(\sin\text{x})^{\tan\text{x}}$
→
Find the intervals in which the following functions are increasing or decreasing.
$f(x) = 6 - 9x - x^2$​​​​​​​
→
Differentiate the following functions with respect to x:
$\text{x}^{\text{x}^2-3}+(\text{x}-3)^{\text{x}^2}$
→
If $\text{y}=(\cos\text{x})^{\cos\text{x}^{\cos\text{x}^{.....\infty}}},$ prove that $\frac{\text{dy}}{\text{dx}}=-\frac{\text{y}^2\tan\text{x}}{(1-\text{y}\log\cos\text{x})}$
→